Sudipta Ray ^1* Sandeep Saha ²

• ¹ Department of Mechanical Engineering, Indian Institute of Technology Kharagpur, Kharagpur, West Bengal, India
• ² Department of Aerospace Engineering, Indian Institute of Technology Kharagpur, Kharagpur, West Bengal, India

The solution of the Poisson equation on multiple domains, separated by an interface, occurs in many physical problems, such as multi-phase flows, propagation of the phase boundary, and diffusion across an interface. The solution embedded with a discontinuity is unique for specified boundary conditions and conditions of discontinuity at the interface. Experimental measurements may be used to aid the solution further and improve our understanding of physical problems with measurement uncertainties. We present a solution of the Poisson equation in a regular Cartesian domain containing a complex-shaped interface, using discrete data from an analytical solution superposed with a disturbance for the interfacial discontinuity as a substitute for measurements.The information for the condition of discontinuity is obtained by smoothly extending the solution in each subdomain over the entire domain. In effect, we acquire the discontinuity data as the difference between the discontinuous parts of the solution, from a limited number of discrete data points, either equally-spaced or variably-spaced, throughout the domain, including the domain boundary, but not necessarily on the interface. A Chebyshev interpolant is constructed using the discontinuity data on the data points, to represent the interface correction function globally. The Chebyshev interpolation allows for fast and exponentially accurate computation of higher-order derivatives of the correction function, which represent discontinuities in the higherorder derivatives of the solution. In the Poisson solver, the discontinuous part of the solution is expressed as a Heaviside function modified with the interface correction function, and the remaining smooth part is solved using Chebyshev polynomials. The Chebyshev collocation method achieves O(10 -14 )-accuracy using N = 20 collocation points in each coordinate direction in two dimensions, in the absence of any superimposed disturbance i.e. for measurement data without experimental uncertainties, with at most M = 225 data points. Furthermore, if the measured interface data is O(ϵ)-accurate due to uncertainties in measurements, the numerical 1 S. Ray, S. Sahaaccuracy also plateaus at O(ϵ), demonstrating the sensitivity of the proposed Chebyshev interpolation method to perturbations in the discontinuity measurements.